Add test MQTT subscription is completed when birth message is sent #188839

Job	Run time
Collect information & changes data	15s
Prepare dependencies (3.12)	12s
Prepare pre-commit base	14s
Check pylint	49s
Check hassfest	32s
Check mypy	21s
Check all requirements	22s
Split tests for full run	0s
Check ruff-format	27s
Check ruff	22s
Check other linters	25s
Run tests Python 3.12 (mqtt)	3m 33s
Run ${{ matrix.postgresql-group }} tests Python ${{ matrix.python-version }}	0s
Run ${{ matrix.mariadb-group }} tests Python ${{ matrix.python-version }}	0s
Run tests Python ${{ matrix.python-version }} (${{ matrix.group }})	0s
Upload test coverage to Codecov (full suite)	0s
Upload test coverage to Codecov (partial suite)	10s
	7m 42s

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